We don't have a way to do that via the POST method, but sellers can export their E-junkie Transaction Log to a zipped, tab-delimited text file, so you might look into a way to facilitate importing that exported data to QuickBooks via your solution.
BTW, note that the initial log view always defaults to show only the last rolling-month of orders, and the Download Log button only exports your current log view at that moment. To export a full order history, the seller would need to select a different date range covering the history they want and click View Log, then click Download Log to export that log view.
